Protein microtubules take part in several cellular activities including mitosis, cell movement and migration. During these
cellular activities, they can be subject to various types of external loading and pressure. In this study, the bucking of protein
microtubules obtained via scale-dependent continuum models are investigated. Several continuum-based formulations, which have
been proposed for the buckling of protein microtubules, are reviewed briefly. Finally, the effects of surface elastic properties on the
growth rate of buckling in protein microtubules are studied.
Keywords: Protein microtubules; Buckling; Axial loading; Size effects
Size effects have a crucial role to play in the statics and
dynamics of various ultra-small structures [1-6]. On the other
hand, the mechanics of nanostructures [7-14] and microstructures
[15-26] is of high importance due to their applications in
different nanomechanical and micromechanical systems such
as Nano sensors and nanoactuators. Therefore, developing sizedependent
mathematical frameworks for analyzing the statics
and dynamics of both nanostructures and microstructures would
provide a useful tool in nanoengineering and microengineering.
Protein microtubules are one of the most important parts of
living cells, which participate in many processes inside cells
[27,28]. For instance, in the process of mitosis, microtubules help
chromosomes to separate and migrate into two opposite positions.
In addition, these filaments provide a reliable pathway for protein
transportation inside cells. In these processes, microtubules are
likely to be subject to various loads such as axial compression. In
this study, the buckling instability of protein microtubules under
axial compressive loads is investigated. Different size-dependent
models of these small-scale structures are also reviewed.
Let us consider a single microtubule of length L, inner radius
Ri and outer radius Ro. The microtubule has a hollow cylindrical
geometry and consists of α and β tubulins, as shown in (Figure 1).
It has been proven that size influences have a significant impact on
the mechanica0000000l behavior at small-scales [29-36]. Since the
inner and outer radii of microtubules are of several nanometers,
the nonlocal theory is mostly used to describe size influences.
The nonlocal theory is an elasticity-based theoretical tool, which
was first utilized by Peddieson et al.  for the deformation of
nanostructures. According to this theory, we have the following
differential equation for the constitutive response of microtubules.
Figure 1: The structure of a protein microtubule.
In which σ , C and ε are, respectively, the stress, elasticity and
strain tensors; moreover, ∇2 and e0lc stand for the Laplace operator
and nonlocal constant, respectively; also, lc and e0 are symbols,
which are used for calibrating the model and incorporating the
effects of the internal configuration of the structure [38,39]. In
addition to nonlocal effects, surface influences have a crucial role to
play in the mechanics of ultra small structures such as microtubules.
At nanoscales, surface influences become important since the ratio
of the surface energy to its bulk counterpart substantially increases.
For the microtubule, there are two different surface layers (i.e.
outer and inner surface layers). To incorporate surface influences,
the following equations can be utilized [40,41].
Here “sur” is employed to indicate “surface”. λsur is the residual
stress in surface layers , and ∧ represents the microtubule
surface energy density. Figure 2 depicts the dimensionless
growth rate of buckling in protein microtubules  subject to
axial compression. Calculations are conducted for various surface
elastic constants . The horizontal axis of the figure denotes
the instability wave number. It is concluded that the growth rate
of buckling in microtubules decreases when the elastic constant of
surface layers increases. This is because of the fact that the surface
elastic constant is associated with an increase in the microtubule
Figure 2: Buckling behaviour of microtubules for different surface elastic constants .
The buckling instability of microtubules in human cells has
been investigated via scale-dependent theoretical models. Two
main scale-dependent theories for the statics and dynamics of
microtubules (i.e. surface and nonlocal theories of elasticity) were
reviewed briefly. Finally, the influences of buckling wave number
and surface elastic constant on the buckling behaviour were studied.
It was concluded that higher surface elastic constants substantially
reduce the growth rate of buckling in the protein microtubule.